Is there any way to avoid Gödel incompleteness theorem?

Question

I was wondering whether there have been successful attempts to avoid somehow the incompleteness theorem. Many point out that every language that is powerful enough to model natural numbers can produce a godel-type sentence. But my point is that it's actually a METAmathematical strategy that Godel uses (as he says in the first pages of his paper). Arithmetic doesn't contain itself the predicate 'provable'. So basically I'm asking: can we avoid such inconsistencies just forbidding to represent any predicate like 'provable' unless moving to a higher-predicate metalanguage?

Answer 1

As Don Thousand and Mauro ALLEGRANZA pointed out, there are ways to circumvent the need for a provability predicate. A systematic method to go about constructing true but unprovable statements would use that every theory has an ordinal strength $\alpha$, then construct a set $S$ which is well ordered, with order type $\beta\ge\alpha$. Then, encode the statement "$S$ is well ordered" into that system. This gives a true but unprovable statement. An example of this can be found at https://arxiv.org/abs/2003.14342. I know of two ways to circumvent Gödel's incompleteness theorem:

1. A weak theory. Gödel's incompleteness theorem only applies to sufficiently strong systems. It is not applicable to systems like Presburger arithmetic and first-order logic.
2. An inconsistent theory. This pretty much goes without saying, see principle of explosion